Finding freq., Wavelentgh, Phase Velocity, and attenuation constant

[VinnyCee]

Homework Statement

Given...

$$v\left(z,\,t\right)\,=\,5\,e^{-\alpha\,z}\,sin\left(4\pi\,\times\,10^9\,t\,-\,20\pi\,z\right)$$

where z is distance (m), find...

(a) Frequency

(b) Wavelength

(c) Phase Velocity

(d) At z = 2m, the amplitude is 1 [V], Find the attenuation constant ($\alpha$).

Homework Equations

$$f\,=\,\frac{1}{T}$$

$$y\left(x,\,t\right)\,=\,A\,cos\left(\frac{2\pi\,t}{T}\,-\,\frac{2\pi\,x}{\lambda}\,+\,\phi_0\right)$$

$$u_p\,=\,f\,\lambda$$

The Attempt at a Solution

(a)

Using the first term ($\frac{2\pi\,t}{T}$) in the argument to the cosine in the general form above...

$$\frac{2\pi}{T}\,=\,4\pi\,\times\,10^9\,\,\longrightarrow\,\,T\,=\,\frac{2\pi}{4\pi\,\times\,10^9}\,=\,0.5\,\times\,10^{-9}$$

$$f\,=\,\frac{1}{T}\,=\,\frac{1}{0.5\,\times\,10^{-9}}\,=\,2\,\times\,10^9\,=\,2\,Ghz$$(b)

Using the second term ($-\,\frac{2\pi\,x}{\lambda}$) in the argument to the cosine in the general form above...

$$\frac{2\pi}{\lambda}\,=\,20\pi\,\,\longrightarrow\,\,\lambda\,=\,\frac{2\pi}{20\pi}\,=\,\frac{1}{10}\,=\,0.1\,m$$(c)

$$u_p\,=\,f\,\lambda\,=\,\left(2\,\times\,10^9\right)\,(0.1)\,=\,200,000,000\,\frac{m}{s}$$(d)

$$1\,=\,5\,e^{-2\,\alpha}\,sin\left(4\pi\,\times\,10^9\,t\,-\,40\pi\right)$$

$$5\,e^{-2\alpha}\,=\,1\,\,\longrightarrow\,\,-2\alpha\,=\,ln\left(\frac{1}{5}\right)\,\,\longrightarrow\,\,\alpha\,=\,0.8047$$

Right?

 

[unplebeian]

How did you eliminate 't' in part d?

 

[HallsofIvy]

There is no "t" in (d). The "attenuation" constant is the rate at which the magnitude of the wave degrades- and that depends entirely upon the coefficient of the cosine term, $5e^{-\alpha z}$. And here, we are given that z= 2.

 

[unplebeian]

What about the sin term. 1= 5xexp(-alpha x z) x sin term which contains t?